Properties

Label 242.2.a
Level $242$
Weight $2$
Character orbit 242.a
Rep. character $\chi_{242}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $6$
Sturm bound $66$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 242.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(66\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(242))\).

Total New Old
Modular forms 45 10 35
Cusp forms 22 10 12
Eisenstein series 23 0 23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(11\)FrickeDim
\(+\)\(+\)$+$\(2\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(4\)
\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(7\)

Trace form

\( 10 q - 2 q^{3} + 10 q^{4} - 2 q^{5} + 8 q^{9} + O(q^{10}) \) \( 10 q - 2 q^{3} + 10 q^{4} - 2 q^{5} + 8 q^{9} - 2 q^{12} - 4 q^{15} + 10 q^{16} - 2 q^{20} - 4 q^{23} + 4 q^{25} - 2 q^{26} - 8 q^{27} - 8 q^{31} - 4 q^{34} + 8 q^{36} - 10 q^{37} - 2 q^{38} - 4 q^{42} - 10 q^{45} - 8 q^{47} - 2 q^{48} + 2 q^{49} - 6 q^{53} - 6 q^{58} - 6 q^{59} - 4 q^{60} + 10 q^{64} - 18 q^{67} - 16 q^{69} - 8 q^{70} - 12 q^{71} - 22 q^{75} - 8 q^{78} - 2 q^{80} - 14 q^{81} - 12 q^{82} - 10 q^{86} - 16 q^{89} + 64 q^{91} - 4 q^{92} + 60 q^{93} + 60 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(242))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 11
242.2.a.a 242.a 1.a $1$ $1.932$ \(\Q\) None \(-1\) \(-2\) \(-3\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-3q^{5}+2q^{6}+2q^{7}+\cdots\)
242.2.a.b 242.a 1.a $1$ $1.932$ \(\Q\) None \(1\) \(-2\) \(-3\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-3q^{5}-2q^{6}-2q^{7}+\cdots\)
242.2.a.c 242.a 1.a $2$ $1.932$ \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
242.2.a.d 242.a 1.a $2$ $1.932$ \(\Q(\sqrt{5}) \) None \(-2\) \(3\) \(2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)
242.2.a.e 242.a 1.a $2$ $1.932$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
242.2.a.f 242.a 1.a $2$ $1.932$ \(\Q(\sqrt{5}) \) None \(2\) \(3\) \(2\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta )q^{3}+q^{4}+(2-2\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(242))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(242)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)